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Chapter 7: Q. 79 (page 605)

Prove the statements about the convergence or divergence of sequences in Exercises 78鈥�83, referring to theorems in the section as necessary. For each of these statements, assume that r is a real number and p is a positive real number.
${(\frac{1}{k})}^{p} 鈫� 鈭�$

Short Answer

Expert verified

Hence proved that $\lim_{k 鈫� 鈭�} {(\frac{1}{k})}^{p} = 0$

Step by step solution

01

Step 1. Given information

The given sequence ${(\frac{1}{k})}^{p} 鈫� 鈭�$

02

Step 2. Prove that 1kp→0

For

p > 0

, for increasingk, the terms of the sequence

\{\frac{1}{k^{p}}\}

decreases. The sequence

\{\frac{1}{k^{p}}\}

is a decreasing sequence.
It is given that

p > 0

, therefore,

\lim_{k 鈫� 鈭�} {(\frac{1}{k})}^{p} = 0

For $蔚 > 0$

$|{(\frac{1}{k})}^{p} - 0| < 蔚$

Therefore exists an Ngreater than ${(\frac{1}{蔚})}^{\frac{1}{p}}$ ,then

$|{(\frac{1}{k})}^{p} - 0| < 蔚 f o r k 鈮� N$

Thus $\{\frac{1}{k^{p}}\}$ is a null sequence

Hence,role="math" localid="1649311817934" $\lim_{k 鈫� 鈭�} {(\frac{1}{k})}^{p} = 0$

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Most popular questions from this chapter

In Exercises 48鈥�51 find all values of p so that the series converges.

${鈭�}_{k = 2}^{鈭�} \frac{1}{k {(\ln (k))}^{p}}$

Prove Theorem 7.24 (a). That is, show that if c is a real number and ${鈭�}_{k = 1}^{鈭�} a_{k}$ is a convergent series, then ${鈭�}_{k = 1}^{鈭�} c a_{k} = c {鈭�}_{k = 1}^{鈭�} a_{k}$ .

Find an example of a continuous function f : $[1, 鈭�) 鈫� R$ such that ${鈭�}_{1}^{鈭�} f (x) d x$ diverges and localid="1649077247585" ${鈭�}_{k = 1}^{鈭�} f (k)$ converges.

In Exercises 48鈥�51 find all values of p so that the series converges.

${鈭�}_{k = 1}^{鈭�} \frac{\ln (k)}{k^{p}}$

Given that $a_{0} = - 3, a_{1} = 5, a_{2} = - 4, a_{3} = 2$ and ${鈭� a_{k}}_{k = 2}^{鈭�} = 7$ , find the value ofrole="math" localid="1648828282417" ${鈭� a_{k}}_{k = 1}^{鈭�}$ .

See all solutions

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